Question

I have an array of “lines” each defined by 2 points. I am working with only the line segments lying between those points. I need to search lines that could continue one another (relative to some angle) and lie on the same line (with some offset)

I mean I had something like 3 lines

I solved some mathematical problem (formulation of which is my question) and got understanding that there are lines that could be called relatively one line (with some angle K and offset J)

And of course by math formulation I meant some kind of math formula like

Answer 1

1. Sort all your segments based on angle (in range 0 to Pi), and build a sorted map of angle-to-segment.
2. Decide on some angle difference threshold below which two segments can be considered parallel. Iterate through your map and for each mapping, consider adjacent mappings on either side of the angle (needs wrap around) which are considered parallel.
3. Within each set of nearly-parallel segments, see if they are “continuations” of each other.

Two segments (A,B) and (C,D) are roughly collinear if all possible pairings of the 4 points are roughly parallel. You can use the same test as above.

Pseudo-code:

Angle(A,B)
    return Atan((B.y-A.y) / (B.x-A.x)) // use atan2 if possible, but needs wrapping

NearlyParallel(angle1, angle2)
    delta = Abs(angle1-angle2)
    return (delta < threshold) or (delta > Pi-threshold)

Collinear(A,B, C,D)
    // Assume NearlyParallel(Angle(A,B), Angle(C,D)) == true
    return NearlyParallel(Angle(A,C), Angle(B,D)) and NearlyParallel(Angle(A,D), Angle(B,C))